SOME RESULTS ON CHARACTERIZATION OF FINITE GROUPS BY NON-COMMUTING GRAPH

Abstract. The non commuting graph r(G) of a non-abelian finite group G is defined as follows: its vertex set is G - Z(G) and two distinct vertices x and y are joined by an edge if and only if the commutator of x and y is not the identity. In this paper we prove some new results about this graph. In particular we will give a new proof of Theorem 3.24 of [A. Abdollahi, S. Akbari, H. R, Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006) 468-492.]. We also prove that if G1;G2; ...... ;Gn are finite groups such that Z(Gi) = 1 for i = 1; 2; ... ; n and they are characterizable by non commuting graph, then G1 * G2 * ..... * Gn is characterizable by non-commuting graph.